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Lecture 9Consumption and SavingsNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Williams Economics 312Household Problemmaxc,c0u(c) + βu(c0) s.t. c +c01 + r= yPVForm Lagrangian with multiplier λ > 0.L = u(c) + βu(c0) + λyPV− c −c01 + rFOC: u0(c) = λβu0(c0) =λ1 + rCombine them to get Euler Equation:u0(c) = β (1 + r) u0(c0)Williams Economics 312Comparative Statics: Income ChangesWhat happens if y, y0or A increases? All matters is yPV.Both c and c0increase (normal goods).If y or A increase, s increases to finance higher c0.Examples: increases in stock market or house prices –“wealth effect”If y0increases, s falls to finance higher current c.Examples: Announced layoffs, changing professions (orcollege majors).Sometimes discuss marginal propensity to consume (MPC).For the log example, MPC out of current income or wealth:c =yPV1 + β∂c∂A=∂c∂y=11 + β> 0Williams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.8-17Figure 8.5 The Effects of an Increase in Current Income for a LenderWilliams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.8-25Figure 8.9 Stock Prices and Consumption of Nondurables and Services, 1985–2006Williams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.8-26Figure 8.10 Scatter Plot of Percentage Deviations from Trend in Consumption of Nondurables and Services Versus Percentage Deviations from Trend in a Stock Price IndexWilliams Economics 312Comparative Statics: Changes in Interest RateIncome effect: if a saver s > 0, then higher interest rateincreases income for given amount of saving. Increasesconsumption in first and second period. If borrower s < 0,then income effect negative.Substitution effect: gross interest rate 1 + r is relative priceof consumption in period 1 to consumption in period 2.Current c becomes more expensive relative to c0. Thisincreases c0and reduces c.Hence: for a saver an increase in r increases c0and mayincrease or decrease c. For a borrower an increase in rreduces c and may increase or decrease c0.Williams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.8-29Figure 8.12 An Increase in the Real Interest Rate for a LenderWilliams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.8-30Figure 8.13 An Increase in the Real Interest Rate for a BorrowerWilliams Economics 312Infinite Horizon ModelNow extend the consumption-savings model from 2 periodsto an infinite horizon. Many of the same implications.Slightly different timing/notation following Wickens.Flow budget constraint: ctconsumption at date t, atassetson hand at start of t. at+1assets chosen at t, carried overto t + 1, rtinterest rate between t − 1 and t, xtincome:ct+ at+1= xt+ (1 + rt)atDerive intertemporal budget constraint, with r0= 0:c0= x0− a1+ a0= x0−c1− x11 + r1−a21 + r1+ a0= x0−c1− x11 + r1−c2− x2(1 + r1)(1 + r2)−a3(1 + r1)(1 + r2)+ a0c0+c11 + r1+c2(1 + r1)(1 + r2)=x0+x11 + r1+x2(1 + r1)(1 + r2)−a3(1 + r1)(1 + r2)+ a0Williams Economics 312Intertemporal Budget ConstraintContinue same process for any horizon T :TXt=0ctQts=0(1 + rs)=TXt=0xtQts=0(1 + rs)+ a0−aT+1QTs=0(1 + rs)For any finite horizon T we would have aT+1= 0. Noreason to save, and more importantly no one would lend.For infinite horizon, need to rule out the possibility ofborrowing forever and never repaying principal.A Ponzi game occurs when agents borrow, repayingexisting debt obligations by borrowing more. We imposethe No Ponzi Game (NPG) restriction:limT→∞aT+1QTs=0(1 + rs)≥ 0This rules out borrowing indefinitely. Household won’twant to have strictly positive assets in limit, so NPG willhold with equality.Williams Economics 312Household Problem: Infinite HorizonUnder the NPG restriction we can take limits as T → ∞:∞Xt=0ctQts=0(1 + rs)=∞Xt=0xtQts=0(1 + rs)+ a0≡ xPVThe household problem is now to choose {ct}∞t=0tomaximize utility subject to the present value budgetconstraint. Single optimization problem, choosing plan forconsumption for entire future.Lagrangian:L =∞Xt=0βtu(ct) + λ xPV−∞Xt=0ctQts=0(1 + rs)!Williams Economics 312Household Problem: Optimality ConditionsFirst order condition for consumption at any dates t, t + 1:βtu0(ct) =λQts=0(1 + rs)βt+1u0(ct+1) =λQt+1s=0(1 + rs)Divide these two equations:u0(ct)βu0(ct+1)=Qt+1s=0(1 + rs)Qts=0(1 + rs)= 1 + rt+1So once again we get the consumption Euler equation:u0(ct) = βu0(ct+1)(1 + rt+1)This governs behavior of consumption for any dates t, t + 1.Williams Economics 312The Life Cycle HypothesisOne application: Franco Modigliani’s life-cycle hypothesisof consumptionIndividuals want smooth consumption profile over theirlife. Labor income varies substantially over lifetime,starting out low, increasing until around the 50th year of aperson’s life and then declining until retirement around 65,with no labor income after retirement.Life-cycle hypothesis: by saving and borrowing individualsturn a very non-smooth labor income profile into a verysmooth consumption profile.Williams Economics 312Life-Cycle Hypothesis: An ExampleSuppose that rt= r ∀t, and β(1 + r) = 1. Then Eulerequation implies ct= ct+1= ¯c.Use present value budget constraint to work outconsumption level:∞Xt=0ct(1 + r)t= xPV⇒ ¯c∞Xt=01(1 + r)t=¯c(1 + r)r= xPVSo ct=r1+rxPVfor all t.If xt=r1+rxPVfor all t then at= 0 for all t.Williams Economics 312Life-Cycle PredictionsIn general, consumption is constant but income xtvaries.How is this implemented?c0= x0− a1+ a0⇒ a1= x0− c0+ a0a1= x0+ a0−r1 + rxPVIf current income x0+ a0is low relative tor1+rxPV, borrowa1< 0.If x0+ a0is high relative tor1+rxPV, save a1> 0.These same general implications extend to varying rt,β(1 + rt) 6= 1.Main predictions: current consumption depends on totallifetime income and initial wealth. Saving should follow avery pronounced life-cycle pattern with borrowing in theearly periods of an economic life, significant saving in thehigh earning years from 35-50 and dissaving in retirementyears.Williams Economics 312Abel/Bernanke, Macroeconomics, © 2001 Addison Wesley Longman, Inc. All rights reservedFigure 4.A.5 Life-cycle consumption, income, and savingWilliams Economics 312Life-Cycle EvidenceThis pattern of life-cycle savings is generally consistentwith the dataOne empirical puzzle: Older household do not dissave tothe extent predicted by the theory. Several explanations:1Individuals are altruistic


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UW-Madison ECON 312 - Consumption and Savings

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